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Properties

Label 2-546-39.5-c1-0-20
Degree $2$
Conductor $546$
Sign $0.314 + 0.949i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (1.65 − 0.506i)3-s + 1.00i·4-s + (−0.645 − 0.645i)5-s + (−1.52 − 0.813i)6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (2.48 − 1.67i)9-s + 0.912i·10-s + (−0.346 + 0.346i)11-s + (0.506 + 1.65i)12-s + (0.964 − 3.47i)13-s − 1.00i·14-s + (−1.39 − 0.742i)15-s − 1.00·16-s + 3.74·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.956 − 0.292i)3-s + 0.500i·4-s + (−0.288 − 0.288i)5-s + (−0.624 − 0.332i)6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (0.829 − 0.559i)9-s + 0.288i·10-s + (−0.104 + 0.104i)11-s + (0.146 + 0.478i)12-s + (0.267 − 0.963i)13-s − 0.267i·14-s + (−0.360 − 0.191i)15-s − 0.250·16-s + 0.908·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26666 - 0.914941i\)
\(L(\frac12)\) \(\approx\) \(1.26666 - 0.914941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-1.65 + 0.506i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-0.964 + 3.47i)T \)
good5 \( 1 + (0.645 + 0.645i)T + 5iT^{2} \)
11 \( 1 + (0.346 - 0.346i)T - 11iT^{2} \)
17 \( 1 - 3.74T + 17T^{2} \)
19 \( 1 + (0.564 - 0.564i)T - 19iT^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 + 4.18iT - 29T^{2} \)
31 \( 1 + (1.21 - 1.21i)T - 31iT^{2} \)
37 \( 1 + (-2.17 - 2.17i)T + 37iT^{2} \)
41 \( 1 + (-2.35 - 2.35i)T + 41iT^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 + (7.22 - 7.22i)T - 47iT^{2} \)
53 \( 1 - 5.03iT - 53T^{2} \)
59 \( 1 + (-2.23 + 2.23i)T - 59iT^{2} \)
61 \( 1 + 4.58T + 61T^{2} \)
67 \( 1 + (-1.55 + 1.55i)T - 67iT^{2} \)
71 \( 1 + (1.98 + 1.98i)T + 71iT^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 + 5.80T + 79T^{2} \)
83 \( 1 + (-1.45 - 1.45i)T + 83iT^{2} \)
89 \( 1 + (12.0 - 12.0i)T - 89iT^{2} \)
97 \( 1 + (-6.69 + 6.69i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43552673767283430484456587209, −9.704743825386261168024546528240, −8.777642590740875886022348417871, −8.053353711945782551038054416746, −7.53412494816810630776430349923, −6.16559023483707076300169095027, −4.71853881032582171385004384580, −3.51331432950576695099121196962, −2.55845027347166772133016630204, −1.12153499312664704917105288152, 1.62244702656999144573228098075, 3.16877114231331012028351707742, 4.25369975757161992814950088080, 5.40026513788696621741498401311, 6.84621583690854594680275731115, 7.45598157483400040995212377352, 8.354222444090648039592373128484, 9.102004867034709552815668980470, 9.884555888513056375327751058072, 10.78890446255641403776671488758

Graph of the $Z$-function along the critical line